I’m glad to hear they’ll pay if they have to. Apply to the Ivies and any other elite schools they (and hopefully you too) like, but they shouldn’t be your only choice as you then run the risk of a complete shut out.
Regarding not liking Rutgers: I know that’s a common view amongst New Jerseyans but it’s really not that bad. If your parents are looking up USNWR rankings, Rutgers is ranked #55 nationally. There are over 2,800 four year colleges in America, which means (at least according to USNWR), that Rutgers is in the top 2%. Perhaps mention this to your parents.